Two generalizations on the minimum Hamming distance of repeated-root constacyclic codes

TL;DR

This paper generalizes the calculation of minimum Hamming distances for certain constacyclic codes of lengths involving powers of primes, including cyclic and negacyclic codes, over finite fields.

Contribution

It extends previous results by providing formulas for the minimum Hamming distance of these codes with new generator polynomials.

Findings

Derived minimum Hamming distances for codes generated by specific polynomials

Generalized results to codes of lengths $np^s$ and $2np^s$

Included special cases for cyclic and negacyclic codes of length $2p^s$

Abstract

We study constacyclic codes, of length $np^s$ and $2np^s$, that are generated by the polynomials $(x^n + \gamma)^{\ell}$ and $(x^n - \xi)^i(x^n + \xi)^j$\ respectively, where $x^n + \gamma$, $x^n - \xi$ and $x^n + \xi$ are irreducible over the alphabet $\F_{p^a}$. We generalize the results of [5], [6] and [7] by computing the minimum Hamming distance of these codes. As a particular case, we determine the minimum Hamming distance of cyclic and negacyclic codes, of length $2p^s$, over a finite field of characteristic $p$.